MAINT: Chain exceptions in several places - #16121

@keremh - a gentle ping to see if you are still interested in pursuing these changes. It looks all that remains is to address comments about changing the tests then this should be ready for re-review. Note the CI failure is unrelated.

This PR seemed to have gotten stuck on some merge conflicts following #16633, so I went ahead and rebased on master to try to get everything back to a reviewable state. I also took the liberty of incorporating the bit of feedback from previous reviews about not modifying the handling of exceptions in the tests.

I did notice one last thing: it seems strange that loadtxt and genfromtxt raise different classes of exception in very similar scenarios... but I don't want to overload this PR too much - we'll see if maybe @eric-wieser or @seberg can give it a final once-over then we can get this in.

Looks good to me, thanks everyone.